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Register a new userComputing Quantum Channel Capacities
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Added by
Navneeth Ramakrishnan
Publication date
2019
fields
Physics
and research's language is
English
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The capacity of noisy quantum channels characterizes the highest rate at which information can be reliably transmitted and it is therefore of practical as well as fundamental importance. Capacities of classical channels are computed using alternating optimization schemes, called Blahut-Arimoto algorithms. In this work, we generalize classical Blahut-Arimoto algorithms to the quantum setting. In particular, we give efficient iterative schemes to compute the capacity of channels with classical input and quantum output, the quantum capacity of less noisy channels, the thermodynamic capacity of quantum channels, as well as the entanglement-assisted capacity of quantum channels. We give rigorous a priori and a posteriori bounds on the estimation error by employing quantum entropy inequalities and demonstrate fast convergence of our algorithms in numerical experiments.
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Communication over a noisy channel is often conducted in a setting in which different input symbols to the channel incur a certain cost. For example, for bosonic quantum channels, the cost associated with an input state is the number of photons, which is proportional to the energy consumed. In such a setting, it is often useful to know the maximum amount of information that can be reliably transmitted per cost incurred. This is known as the capacity per unit cost. In this paper, we generalize the capacity per unit cost to various communication tasks involving a quantum channel such as classical communication, entanglement-assisted classical communication, private communication, and quantum communication. For each task, we define the corresponding capacity per unit cost and derive a formula for it analogous to that of the usual capacity. Furthermore, for the special and natural case in which there is a zero-cost state, we obtain expressions in terms of an optimized relative entropy involving the zero-cost state. For each communication task, we construct an explicit pulse-position-modulation coding scheme that achieves the capacity per unit cost. Finally, we compute capacities per unit cost for various bosonic Gaussian channels and introduce the notion of a blocklength constraint as a proposed solution to the long-standing issue of infinite capacities per unit cost. This motivates the idea of a blocklength-cost duality, on which we elaborate in depth.
We analyze a task in which classical and quantum messages are simultaneously communicated via a noisy quantum channel, assisted with a limited amount of shared entanglement. We derive the direct and converse bounds for the one-shot capacity region. The bounds are represented in terms of the smooth conditional entropies and the error tolerance, and coincide in the asymptotic limit of infinitely many uses of the channel. The direct and converse bounds for various communication tasks are obtained as corollaries, both for one-shot and asymptotic scenarios. The proof is based on the randomized partial decoupling theorem, which is a generalization of the decoupling theorem. Thereby we provide a unified decoupling approach to the one-shot quantum channel coding, by fully incorporating classical communication, quantum communication and shared entanglement.
We study information transmission over a fully correlated amplitude damping channel acting on two qubits. We derive the single-shot classical channel capacity and show that entanglement is needed to achieve the channel best performance. We discuss the degradability properties of the channel and evaluate the quantum capacity for any value of the noise parameter. We finally compute the entanglement-assisted classical channel capacity.
We calculate the entanglement-assisted and unassisted channel capacities of an exactly solvable spin star system, which models the quantum dephasing channel. The capacities for this non-Markovian model exhibit a strong dependence on the coupling strengths of the bath spins with the system, the bath temperature, and the number of bath spins. For equal couplings and bath frequencies, the channel becomes periodically noiseless.
We discuss quantum capacities for two types of entanglement networks: $mathcal{Q}$ for the quantum repeater network with free classical communication, and $mathcal{R}$ for the tensor network as the rank of the linear operation represented by the tensor network. We find that $mathcal{Q}$ always equals $mathcal{R}$ in the regularized case for the samenetwork graph. However, the relationships between the corresponding one-shot capacities $mathcal{Q}_1$ and $mathcal{R}_1$ are more complicated, and the min-cut upper bound is in general not achievable. We show that the tensor network can be viewed as a stochastic protocol with the quantum repeater network, such that $mathcal{R}_1$ is a natural upper bound of $mathcal{Q}_1$. We analyze the possible gap between $mathcal{R}_1$ and $mathcal{Q}_1$ for certain networks, and compare them with the one-shot classical capacity of the corresponding classical network.
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